Quick introduction to tensor analysis- sharipov

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MSC 97U20 PACS 01.30.Pp R. A. Sharipov. Quick Introduction to Tensor Analysis: lecture notes. Freely distributed on-line. Is free for individual use and educational purposes. Any commercial use without written consent from the author is prohibited. This book was written as lecture notes for classes that I taught to undergraduate students majoring in physics in February 2004 during mytime as a guest instructor at The University of Akron, which was supported by Dr. Sergei F. Lyuksyutov’s grant from the National Research Council under the COBASE program. These 4 classes have been taught in the frame of a regular Electromagnetism course as an introduction to tensorial methods. I wrote this book in a ”do-it-yourself” style so that I give only a draft of tensor theory, whichincludes formulating definitions and theorems and giving basic ideas and formulas. All other work such as proving consistence of definitions, deriving formulas, proving theorems or completing details to proofs is left to the reader in the form of numerous exercises. I hope that this style makes learning the subject really quick and more effective for understanding and memorizing. I am grateful toDepartment Chair Prof. Robert R. Mallik for the opportunity to teach classes and thus to be involved fully in the atmosphere of an American university. I am also grateful to Mr. M. Boiwka (mboiwka@hotmail.com) Mr. A. Calabrese (ajc10@uakron.edu) Mr. J. Comer (funnybef@lycos.com) Mr. A. Mozinski (arm5@uakron.edu) Mr. M. J. Shepard (sheppp2000@yahoo.com) for attending my classes and reading themanuscript of this book. I would like to especially acknowledge and thank Mr. Jeff Comer for correcting the grammar and wording in it. Contacts to author. Office: Mathematics Department, Bashkir State University, 32 Frunze street, 450074 Ufa, Russia Phone: 7-(3472)-23-67-18 Fax: 7-(3472)-23-67-74 Home: 5 Rabochaya street, 450003 Ufa, Russia Phone: 7-(917)-75-55-786 E-mails: R Sharipov@ic.bashedu.ru,r-sharipov@mail.ru, ra sharipov@hotmail.com, URL: http:/ /www.geocities.com/r-sharipov

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CONTENTS.

CONTENTS. ............................................................................................... 3. CHAPTER I. PRELIMINARY INFORMATION. .......................................... 4. § § § § § § § 1. 2. 3. 4. 5. 6. 7. Geometrical and physical vectors............................................................ 4. Bound vectors and free vectors. .............................................................. 5. Euclidean space. .................................................................................... 8. Bases and Cartesian coordinates. ........................................................... 8. What if we need to change a basis ?...................................................... 12. What happens to vectors when we change the basis ? ............................ 15. What is the novelty about vectors that we learned knowing transformation formula for their coordinates ? ....................................... 17.

CHAPTER II. TENSORS IN CARTESIAN COORDINATES. ..................... 18. § § § § § § § § § § § 8.Covectors. ........................................................................................... 9. Scalar product of vector and covector. .................................................. 10. Linear operators. ............................................................................... 11. Bilinear and quadratic forms. ............................................................. 12.General definition of tensors. .............................................................. 13. Dot product and metric tensor. .......................................................... 14. Multiplication by numbers and addition. ............................................. 15. Tensor product. ................................................................................. 16. Contraction....